(Time–)Frequency Analysis of EEG Waveforms
Niko Busch
Charité University Medicine Berlin; Berlin School of Mind and Brain
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From ERP waveforms to waves
ERP analysis:
time domain analysis: when do things (amplitudes) happen?
treats peaks and troughs as single events.
Frequency domain (spectral) analysis (Fourier analysis):
magnitudes and frequencies of waves — no time information.
peaks and troughs are not treated as separate entities.
Time–frequency analysis (wavelet analysis):
when do which frequencies occur?
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2 / 23
From ERP waveforms to waves
ERP analysis:
time domain analysis: when do things (amplitudes) happen?
treats peaks and troughs as single events.
Frequency domain (spectral) analysis (Fourier analysis):
magnitudes and frequencies of waves — no time information.
peaks and troughs are not treated as separate entities.
Time–frequency analysis (wavelet analysis):
when do which frequencies occur?
[email protected]
2 / 23
From ERP waveforms to waves
ERP analysis:
time domain analysis: when do things (amplitudes) happen?
treats peaks and troughs as single events.
Frequency domain (spectral) analysis (Fourier analysis):
magnitudes and frequencies of waves — no time information.
peaks and troughs are not treated as separate entities.
Time–frequency analysis (wavelet analysis):
when do which frequencies occur?
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2 / 23
Why bother?
(Time–)Frequency analysis complements signal analysis:
neurons are oscillating.
analysis of signals with trial-to-trial jitter.
analysis of longer time periods.
analysis of pre-stimulus and spontaneous signals.
necessary for sophisticated methods (coherence, coupling, causality, etc.).
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Parameters of waves
Oscillations regular repetition of some measure over several cycles.
Wavelength length of a single cycle (a.k.a. period).
1
— the speed of change.
Frequency wavelength
Phase current state of the oscillation — angle on the unit circle. Runs
from 0◦ (-π) — 360◦ (π)
Magnitude (permanent) strength of the oscillation.
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How to disentangle oscillations
Jean Joseph Fourier (1768—1830): An arbitrary function, continuous or with
”
discontinuities, defined in a finite interval by an arbitrarily capricious graph can
always be expressed as a sum of sinusoids“.
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The discrete Fourier transform
The DFT transforms the signal from the time domain into the frequency
domain.
Requires that the signal be stationary.
5 Hz
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The discrete Fourier transform
The DFT transforms the signal from the time domain into the frequency
domain.
Requires that the signal be stationary.
EEG raw data
reverted raw data
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The discrete Fourier transform
The DFT transforms the signal from the time domain into the frequency
domain.
Requires that the signal be stationary.
Non-stationary signals:
When does the 10 Hz oscillation occur?
DFT does not give time information.
Time information is not necessary for stationary signals
Frequency contents do not change — all frequency components exist all the
time.
How to investigate event–related spectral changes in brain signals?
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Event–related synchronisation / desynchronisation
Cut the signal in two time windows and assume stationarity in each half.
ERD/ERS 1 =
poststimulus power −baseline power
baseline power
∗ 100
But why not use even smaller windows?
⇒ Windowed FFT / Short term Fourier transform.
1 Pfurtscheller
& Lopes da Silva (1999). Clin Neurophysiol
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The short term Fourier transform (STFT) I
Assume that some portion of a non–stationary signal is stationary.
Important parameters:
window function (Hamming, Hanning, Rectangular, etc.)
window overlap
window length: width should correspond to the segment of the signal where its
stationarity is valid.
1.5
Hanning window
Hamming window
Boxcar window
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The short term Fourier transform (STFT) I
Assume that some portion of a non–stationary signal is stationary.
Important parameters:
window function (Hamming, Hanning, Rectangular, etc.)
window overlap
window length: width should correspond to the segment of the signal where its
stationarity is valid.
EEG raw data
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shifted Hanning window
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windowed EEG signal
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The short term Fourier transform (STFT) I
Assume that some portion of a non–stationary signal is stationary.
Important parameters:
window function (Hamming, Hanning, Rectangular, etc.)
window overlap
window length: width should correspond to the segment of the signal where its
stationarity is valid.
EEG raw data
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SPECTROGRAM, R = 256
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frequency
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time
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The short term Fourier transform (STFT) II
Window length affects resolution in time and frequency
short window: good time resolution, poor frequency resolution.
long window: good frequency resolution, poor time resolution.
EEG raw data
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Uncertainty principle
Werner Heisenberg (1901—1976):
Energy and location of a particle cannot be both known with infinite precision.
a result of the wave properties of particles (not the measurement).
Applies also to time–frequency analysis:
We cannot know what spectral component exists at any given time instant.
What spectral components exist at any given interval of time?
Spectral/temporal resolution trade off cannot be avoided — but it can be
optimised.
Good frequency resolution at low frequencies.
Good time resolution at high frequencies.
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From STFT to wavelets
STFT: fixed temporal & spectral resolution
Analysis of high frequencies ⇒ insufficient temporal resolution.
Analysis of low frequencies ⇒ insufficient spectral resolution.
Wavelet analysis:
Analysis of high frequencies ⇒ narrow time window for better time resolution.
Analysis of low frequencies ⇒ wide time window for better spectral resolution.
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What is a wavelet?
Motherwavelet
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|ejω0t|
e−t /2
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Wavelet
Zero mean amplitude.
Finite duration.
Mother wavelet: prototype function (f = sampling frequency).
Wavelets can be scaled (compressed) and translated.
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What is a wavelet?
Motherwavelet
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Spectral Density
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Zero mean amplitude.
Finite duration.
Mother wavelet: prototype function (f = sampling frequency).
Wavelets can be scaled (compressed) and translated.
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Wavelet transform of ERPs
Bandpass–filtered ERP
ERP
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Gamma–Band: ca. 30 – 80 Hz
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... but be careful:
any signal can be represented as oscillations w. time-frequency analysis
but it does not imply that the signal is oscillatory!
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Important parameters of a wavelet
B Wavelet - frequency domain
A Wavelet - time domain
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Amplitude
Amplitude
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Length how many cycles does a wavelet have?
e.g. 40 Hz wavelet (25 ms/cycle), 12 cycles ⇒ 250 ms length
σt standard deviation in time domain: σt =
m
2π∗f0
σf standard deviation in frequency domain: σf =
1
2π∗σt
time resolution increases with frequency, whereas frequency
resolution decreases with frequency.
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Evoked and induced oscillations I
evoked/ phase-locked
induced/ non-phase-locked
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Evoked time-frequency representation of the average of all trials (ERP).
Induced average of time-frequency transforms of single trials.
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Evoked and induced oscillations II
Wavelet analysis of single trials reveals non–phase–locked activity.
Evoked/ phase-locked
Induced/ non/phase-locked
Wavelet-transformed trials
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Average
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Evoked time-frequency representation of the average of all trials (ERP).
Induced average of time-frequency transforms of single trials.
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Phase–locking factor (PLF)
a.k.a. intertrial coherence (ITC) or phase–locking–value (PLV).
measures phase consistency of a frequency at a particular time across trials.
PLF = 1: perfect phase alignment.
PLF = 0: random phase distribution.
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The EEG state space
Frequency x phase locking x amplitude changes2
Evoked and induced activity are extremes on a continuum.
ERPs cover only small part of the EEG space.
2 Makeig,
Debener, Onton, Delorme (2004). TICS
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Examples 1: spontaneous EEG
Stimulus pairs are presented at different phases of the alpha rhythm —
sequential or simultaneous?3
If the stimulus pair falls within the same alpha cycle ⇒ perceived simultaneity.
Does the visual system take snapshots at a rate of 10 Hz?
Simultaneity and the alpha rhythm
Figure from VanRullen & Koch (2003): Is perception discrete of continuous? TICS
3 Varela
et al. (1981): Perceptual framing and cortical alpha rhythm.
Neuropsychologia
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Examples 2: pre-stimulus EEG power
Spatial attention to left or right.
Stronger alpha power over ipsilateral hemisphere.4
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Attention: ipsi- vs. contra-lateral
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4 Busch & VanRullen (2010): Spontaneous EEG oscillations reveal periodic sampling
of visual attention. PNAS.
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Examples 3: analysis of long time intervals
Sternberg memory task with different set sizes.
Alpha power increases linearly with set size.5
Effect of set size
5 Jensen et al. (2002): Oscillations in the alpha band (9–12 Hz) increase with
memory load during retention in a short-term memory task. Cereb Cortex.
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Recommended reading
WWW:
EEGLAB’s time-frequency functions explained: http://bishoptechbits.blogspot.com/
FFT explained: http://blinkdagger.com/matlab/matlab-introductory-fft-tutorial/
Wavelet tutorial http://users.rowan.edu/ polikar/WAVELETS/WTtutorial.html
Books:
Barbara Burke Hubbard: The World According to Wavelets.
Steven Smith: The Scientist & Engineer’s Guide to Digital Signal Processing
(http://www.dspguide.com/).
Herrmann, Grigutsch & Busch: EEG oscillations and wavelet analysis. In:
Event-related Potentials: A Methods Handbook.
Papers:
Tallon-Baudry & Bertrand (1999) Oscillatory gamma activity in humans and its role
in object representation. TICS.
Samar, Bopardikar, Rao & Swartz (1999) Wavelet analysis of neuroelectric
waveforms: a conceptual tutorial. Brain Lang.
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Thank you...
... for your interest !
Please ask questions!!!
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